For starters, this 50-minute video pulls together what we've investigated of Riemann surfaces and Dirichlet's principle, and introduces the concept of metaphor:
Riemann has given us a language and a geometry to conceptualize and work with the fundamentally non-quantitative changes seen in human economy, and biology. The goal of this web presentation has been to provide the reader with improved tools to represent non-continuous change as substance, rather than exception (as the book Limits to Growth treated breakthroughs). This allows us to solve the problem facing economic modellers: direct representation of such shifts. We’ll review the content of this presentation so far, and offer some final thoughts on the application of the ideas presented here.
While two things that differ by number only, having a common origin or generating principle, are different quantitatively, the truly exciting changes in evolution and economy are of the character of “more than,” rather than “more of.” This necessarily faces anyone studying human economy, where quantitative measures (e.g., money) fail to represent the qualitatively different economy as a whole that is created by the incorporation of new physical principles into human practice. For example, the Apollo moon mission was said to have had a payback rate of 14 dollars returned for every dollar invested. While the exact numerical ratio might be debated, there are two obvious problems with such a calculation. First, the investment is made once, while the new payback technologies (as concepts, rather than objects) last forever. Secondly, the “payback” dollars aren’t the same as the investment dollars, since the economies in which the dollars existed differed in the same qualitative way as the possibilities of a five-axis mill compared to a three-axis mill. The pre-Apollo cost of various medical technologies was not 14 times higher than the post-Apollo cost; rather, the technologies simply did not exist. This is the peculiar characteristic of economic science: when creative improvements are fostered, the payback cannot be measured in terms of the cost.
LaRouche has used the concept of “energy flux density” to make this distinction clear, by drawing attention not to the quantity of energy, but its density – its quality. While the measured intensity of energy does increase in the shifts from wood-burning, to coal, to petroleum, to nuclear fission, to fusion, and anti-matter, it must be remarked that when measuring the density of energy (which is increasing), the energy itself is not the same. That is, while you could measure the horsepower of a coal engine to compare it to the simple mechanical work that could be performed by a horse, no quantity of horses (or coal) could power an airplane. The transmutations effected in a fission process could not be created by a hydroelectric plant of equivalent electrical power.
There are many excellent geometric examples that allow us to compare higher generating powers with similar projected characteristics. For example, the side of a square with double the area of a given square cannot be created as a length, but only by an area construction. Taking a length of one, and dividing it, adding to it, etc., will not create the side of a square of two, but starting with a square, it is possible to make a new square of doubled area. The doubled square does have a 1-dimensional side length, but it is a length that could only be created in a 2-d construction. The length is two-dimensional in its origin, although not in its appearance.
Similarly, contrasting the rotation along the round circumference of the circle with the motion along its diameter, produces the sine and cosine relationships, which transcend expression by algebra. There is no way to express the true roundness of the circle, from the standpoint of straight lines. There is a new quality in the circle, that cannot be created quantitatively from straightness.
Higher transcendentals, those discovered by Abel and treated by Riemann, continue as a series of increasing density. This perfectly matches the healthy development of human economy, where later stages do not stand in numerical relation to what came before. This is also seen in biology: what's the numerical relation between a larval caterpillar and an adult butterfly?
Connectedness / Topology
We can draw a distinction between the way a process behaves locally, and the global environment in which that activity unfolds. It is possible to move in a “straight,” shortest line on a plane, a sphere, and a torus. The local behavior of simple extension is the same in all three cases, but the global characteristics of the surface itself, considered as an entirety, are different.
Such topological differences show up in the connectedness of surfaces, in the varying number and type of connections between regions. For example, a torus has two independent types of periodic motion: two independent, new means of connecting its parts. Although drawing a very direct analogy can lead away from the real point, there are many examples of this in life and economy.
If we compare the development of multicellular life, we see several stages. First, are single, independent cells such as today’s bacteria. Slime molds present a slightly higher stage, where a number of identical cells form a large plasmodium, a large mass of what are almost independent cells, but without cell membranes separating them. In this stage, the slime mold is like a single cell, while also being similar to multicellular life. In higher multicellular life (animals, plants, and fungi), the different cells making up the organism display very different properties (compare a muscle cell and a skin cell, or a root and a leaf). This requires chemical, electromagnetic, and possibly other fields and signalling/coordinating mechanisms to keep the different tissue types performing their specific tasks. These stages show distinctly different qualities of connections between the constituent parts: they reveal new types of what a “whole” can be in life.
With larger organisms come more opportunities for interaction between what are officially different species. Legumes, for example, have nodules on their roots, full of nitrogen-fixing bacteria. These bacterial cells are not part of the plant itself, yet they function as though they were a specialized organ. Parasites living within or upon other organisms display a new type of connection between different lifeforms, blurring the distinctions between different organisms. On a smaller scale, endosymbiosis was responsible for the chloroplasts in modern plants, which were once free-living photosynthetic unicellular organisms, but have become, like animal mitochondria (which have their own DNA), permanent internal inhabitants. Higher stages of life are characterized by an increasing density of possible interrelations among life, indicating a more complex global geometry of the biosphere as a whole.
Railroads transformed the productive economic potential of the United States in more ways than you might imagine. Obviously, they moved things around the country more quickly, but they did more than that. If you take the midwest, for example, including Detroit, railroads allowed for totally new types of production. Because it was so much easier to move goods from one place to another, it became possible to have more specialized manufacturing, rather than factories that made all the components for their finished goods. The railroad, along with drafting skills and uniformity of measurement, brought into existence a type of economic functioning in the area that had not existed before. Not only goods, but people could travel more quickly, allowing the more rapid spreading of ideas and improved collaboration among thinkers. The railroad didn't simply move already-existing goods and people more quickly; it made previously impossible trips possible, and allowed the creation of new goods that could not have been created otherwise, through a new type of connectedness.
The economic reforms of Jean-Baptiste Colbert serve as another example. In making France truly a sovereign nation-state, with control over its internal financial policies, Colbert changed the geometry. Before his reforms, each duke or other noble in control of a piece of land or a segment of a river was free to charge whatever taxes he liked on goods entering or leaving his fiefdom. While this had a local profit-maximizing effect in the minds of the nobles, it significantly hampered trade within France as a whole. By cracking down on such taxes, by such means as putting various goods under the royal backing of the king, Colbert unified the nation, allowing his pro-industry development policies to flourish.
All industrial production, if traced up the line of goods, depends on raw materials. This brings a geographical specificity into economics, since such raw materials are not uniformly distributed around the globe. Just as the advent of agriculture made it possible to change the suitability of land to human habitation, so too will the powerful materials recycling technologies of the future change our view of raw materials sources. Fusion power, in addition to producing electricity, could also supply levels of heat sufficient to vaporize all materials into their elemental components. While this technology may not be energetically useful on earth, it will be essential for space-faring settlements, simplifying mining and materials sourcing to a degree comparable to that introduced by using electrolysis, rather than chemical means, for refining metal ores. The materials cycle will become much shorter, more local, and more densely connected.
I’d like to finish with an example of temporal, rather than spatial connectivity. The dynamic laws governing non-living processes have instantaneous expressions that involve time, but no direction to that time. That is, such physical laws as force interaction, magnetism, and the least-time propagation of light, work just as well if you reverse time for the unfolding process as a whole. For living processes, time most certainly is directed, both in the life of an individual organism, as well as evolutionary time. For human beings, this time takes on a whole new meaning.
Think first of standard clock-time. In such time, before and after are opposites, but only quantitative opposites. That is, after is not-before in the same way that left is not-right. The dynamic laws of physics work perfectly well in either direction of time. Now, think of evolutionary time, where later stages of development (say warm-blooded mammals) are not a future extrapolation of a past state. That is, taking past states of life as systems and projecting their future growth, would not create fundamentally new types of life. Here, before and after differ in a more than quantitative way: they are not simply opposites. "After" refers to a state in which a new concept has been introduced. The same goes for real music. While some musical works would sound just as well played backwards, the same is not the case for a piece which develops, where the future does not come out of the past, but rather required it for its generation.
When we introduce new discoveries to society, we have created “before” and “after” states, where the temporal difference is not quantitative, but qualitative. But the human ability to create such shifts takes us as a species outside of time altogether! Our mortal lives occupy a certain number of years, rarely exceeding a century. Yet, we are able to be as efficiently connected to the dead, and the unborn future to us, as we are able to be connected to those alive today, and in many cases, closer. The scientific method of Kepler, which he expresses in breathtaking detail in his major works, can reach our mind in such a way that we have experienced in our minds, the same thoughts and breakthroughs that took place in his. We can know his passion and his intensity. In our present actions, we are able, unlike animals, to be in dialogue with the future we seek to bring about, to talk to people not yet born, as we reflect on how they will reflect on our thoughts and actions.
When you get a new idea, or realize a gross error in thinking, it is true, in many respects, that you really aren't the same person anymore: that person no longer exists. Yet, there is something potentially continuous through your life, and beyond it. That continuous thread, in which all of eternity exists as a single moment, gives an immediate, proximate temporal connection across human history. The physical economic decay around us must be remedied, but the economic growth we must create is not to have more of the same people we have today, just with better living standards. Beyond the physical quality of life, there is also the question of one's mental well-being and clear knowledge of purpose in existing, the knowledge that they, through their lives, have done something that in retrospect will clearly have been necessary. Such is the purpose of national government: to afford its citizens the opportunity to participate in the ongoing development characteristic of human life uniquely among living things, and characteristic of the development of life as a whole, to sing, quoting Kepler, “the perpetuity of the whole of cosmic time in some brief fraction of an hour, by the artificial concert of several voices, and taste up to a point the satisfaction of God his Maker in His works.”
Just as for Kepler, we must look to the stars for this, but, going beyond him, we must also reach them. Space colonization, and NAWAPA, the terra-forming project here on earth that treats our planet experimentally, are the most concentrated expression of the potential for growth of the human species.